Magneto-hydrodynamic peristaltic flow of a Jeffery fluid in the presence of heat transfer through a porous medium in an asymmetric channel

In the present paper, the effects of magnetic field and heat transfer on the peristaltic flow of a Jeffery fluid through a porous medium in an asymmetric channel have been studied. The governing non-linear partial differential equations representing the flow model are transmuted into linear ones by employing the appropriate non-dimensional parameters under the assumption of long wavelength and low Reynolds number. Exact solutions are presented for the stream function, pressure gradient, and temperature. The frictional force and pressure rise are both computed using numerical integration. Using MATLAB R2023a software, a parametric analysis is performed, and the resulting data is represented graphically. For all physical quantities considered, numerical calculations were made and represented graphically. Trapping phenomena are discussed graphically. The obtained results can be applied to enhance pumping systems in engineering and gastrointestinal functions. This analysis permits body fluids such as blood and lymph to easily move inside the arteries and veins, allowing oxygen supply, waste elimination, and other necessary elements.


List of symbols
where the higher and lower waves' amplitudes, respectively, are γ 1 , γ 2 , λ is the wavelength, t time, φ is the phase difference belongs to [0, π].Furthermore, α 1 , α 2 , γ 1 , γ 2 and φ satisfy the inequality below: The governing equations in the laboratory frame are 7,8 The relationships between the two frames are defined as follows: where (u, v), (U, V ), (p, P) and T represent the components of velocity in the wave frame of reference, the velocity components in laboratory frame, the pressure in wave and fixed frame and, the temperature respectively.Since a small Reynolds number is assumed, the induced magnetic field is ignored.In the above equations k is the permeability of the porous medium, Q• is the heat source, 1 is the ratio of relaxation to retardation times, 2 is the retardation time, ρ is the density, K 1 is the thermal conductivity, σ is the electrical conductivity of the fluid, c p is the specific heat, q is the radiative heat flux and ǫ is the porosity of the porous medium and B• is the applied magnetic field.The non-dimensional quantities are given below: Applying (9) in ( 1) and ( 4)- (8) and illuminate the bars, we have, (4)  11), ( 12) and ( 13) and by ignoring terms containing δ and its higher powers by utilizing the long wavelength approximation ( δ << 1) and low Reynolds number assumption, we have and According to Eq. ( 16), p is not dependent on y .Consequently, (15) may be expressed as These non-dimensional boundary conditions are 40,41 equivalent to According to the wave frame, the volumetric flow rate in the non-dimensional form is

Solution of the problem
We get θ and u by solving (15), (17), and applying the boundary conditions (19).
It follows from ( 22) and ( 20) that the pressure gradient can be written as ( 14)

Numerical procedure
Using the command DSlove in the Mathematica program, linear Eqs. ( 17) and ( 18) were solved with boundary conditions (19).This procedure is useful in reducing CPU per evaluation as well as reducing error.Also, to obtain graphical solutions and numerical calculations, an appropriate algorithm has been developed for this matter.

Numerical results and discussion
This section discusses the graphical data that were found in this study for temperature, velocity, pressure gradient, pressure rise, friction forces, stress and streamlines.MATLAB is used for the simulation and the results are exhibited through graphs.For numerical computations, the material properties of parameter values as in Reddy 40 are taken under consideration.Figure 2 depict the impact of thermal radiation and heat source/sink on temperature with respect to y-axis, while keeping other parameters fixed.As increasing the parameter β the temperature increases, while it decreases when the value of the parameter R increasing.The temperature satisfies the boundary conditions.This result is in good agreement with the results obtained by Hayat et al. 6 .Figure 3 displays the change in velocity u with the y-axis according to ǫ is the porosity of the porous medium, thermal radiation R , Gr Grashof number, M Hartmann number, heat source/sink β and Darcy's number Da .It can be seen that increasing Da, β, Gr and ǫ increases the velocity u , while rising R and M cause it to decrease.Moreover, it has the highest value in the channel's middle and the lowest value at the channel's edges.Also, it satisfies the boundary conditions.This is in good agreement with what was obtained in clinical practice because the nutrients diffuse out of the blood vessels to neighboring tissues 23 .
Figure 4 illustrates the variations of the pressure gradient dp dx with respect to the x-axis for various parameters Da, β, Gr, ǫ, M and the phase difference φ .According to the graph, the pressure gradient rises when M, ǫ, Gr are increased while falling when Da, β, φ are decreased.it is observed that the pressure gradient oscillates in the whole range x.For more authenticity, this result is in good agreement with the results obtained by Reddy 40 .
−dp dx dx. ( Changes of β, R on the temperature θ against y-axis. Figure 5 presents the impact of Da, β, Gr, ǫ, M and φ on the pressure rise p with respect to rate volume flow F. It is noticed that the pressure rise decreases with increasing β, Gr, ǫ , while it increases with increasing φ , as well, it increases with increasing M in the region (− 200 ≤ F ≤ 0) and it decreases in the interval (0 ≤ F ≤ 200), otherwise it falls with rising Da in the period (− 200 ≤ F ≤ 0) and increases in the interval (0 ≤ F ≤ 200).This result is in good agreement with the results obtained by Reddy 40 .
Figures 6 and 7 demonstrate how the friction force for upper F u and lower F L varies with respect to a flow rate in volume F under these parameters Da, β, Gr, ǫ, M .It is noticed that the friction force for lower and upper increases with increasing β, Gr, ǫ , as well, they increase with increasing Da in the interval (− 200 ≤ F ≤ 0) and decrease in the interval (0 ≤ F ≤ 200).But by increasing M , they decrease in the interval (− 200 ≤ F ≤ 0) and increase in the interval (0 ≤ F ≤ 200).The behavior of pressure rise is observed to be the inverse of the behavior of friction forces in the upper and lower layers.Figure 8 shows the variations of the shear stress τ xy with respect to x-axis for different values of ǫ, Gr, M, β, R and Da .The stress decreases in the interval ( −0.5 ≤ x ≤ 0 ) and increases in the interval ( 0 ≤ x ≤ 0.5 ) by rising Gr, Da, and β .While, it increases in the interval ( −0.5 ≤ x ≤ 0 ) www.nature.com/scientificreports/and decreases in the interval ( 0 ≤ x ≤ 0.5 ) by rising g ǫ, R, M .Figure 9 shows the 3D schematics concern θ, u, and dp dx , with regards to x and y axes under the effect of heat source β , Hartman number M , and Darcy number Da .The temperature is observed to drop as R increases, while the velocity increases with increasing Da and decreases with increasing M .Also, the pressure gradient decreases with increasing β .In 3D, all physical quantities derived from peristaltic flow overlap and dampen as they increase in order for particles to reach equilibrium.Most physical fields move in peristaltic flow, which is more relevant for the vertical distance of the curves that were created.www.nature.com/scientificreports/with increasing Gr .Also, we found that as M is raised, the trapped bolus's size grows, as shown in Fig. 12a-d.This increase in the size describes the volume of the fluid that is bounded by invariant closed streamlines.Furthermore, compared to the symmetric channel, the size of the trapped bolus is less in the asymmetric channel.www.nature.com/scientificreports/

Conclusion and future work
The role of the magnetic field and heat transfer have been studied for a fluid that is not Newtonian in a porous medium.To solve the problem mathematically, small Reynolds number and long wavelength assumptions are utilized.Graphical illustrations have been used to explain and discuss the impact of different physical factors on the flow characteristics.Below is a list of the important results.
• The partial differential equations that appear in this paper have an accurate analytical solution approach.
• Temperature can be increased by increasing β and reduced by increasing R.
• The flow has a maximum velocity in the centre and subsequently increases with increasing β, Da, Gr, ǫ and drops with increasing M, R , according to the graphical solutions for the velocity.• The magnitude of pressure gradient has an oscillating behaviour as it increases by increments of M, Gr, ǫ and decreases by increments of Da, φ. • Parameters Gr, ǫ , β have a decreasing effect on the pressure rise, while parameter φ has an increasing effect on it, as well, it increases with increasing M in the interval − 200 ≤ F ≤ 0 and decreases in the interval 0 ≤ F ≤ 200, otherwise it drops with raising Da in the period − 200 ≤ F ≤ 0 and increases in the interval 0 ≤ F ≤ 200.• When compared to the pressure rise, the frictional force similarly has the opposite trend.
• The volume of the trapped bolus increases as the magnetic field and Grashof number increase, while it decreases as the heat source increases.• Researchers working in the domains of science, engineering, medicine, and fluid mechanics may find the study's findings beneficial.• Future research may be done in this approach to examine how slip circumstances affect flow characteristics.

Figure 1 .
Figure 1.Diagrammatic depiction of the physical model.

Figure 3 .
Figure 3. Changes of Da, β, Grǫ, M, R on the velocity distribution u against y-axis.

Figure 7 .
Figure 7. Changes of Da, β, Grǫ, M on the friction force lower F L against F.

Figure 9 .
Figure 9. 3D plot of temperature θ, velocity u and the pressure gradient dp dx with x, y axes for changes in β, M and Da respectively.